Somebody else here said "The three roots will be the three dimensions of the box", but that doesn't make any sense. We're told that this function is the prism's volume. How could a prism with non-zero dimensions give us a volume of 0?
The volume of a rectangular prism is always length times width times height. So the product of those has to equal x^3 + 9x^2 + 6x - 16. However, there are lots of measurements whose product might be equal to this. For example, it could be that the length is 1 unit, the width is 1 unit, and the height is x^3 + 9x^2 + 6x - 16 for some unknown x. You could factor it into (x-1)(x+2)(x+8), but again, the length doesn't have to necessarily be one of these. If this expression is a volume of the function, then it begs the question of what exactly x is supposed to measure.
If it is a rectangular prism the area is the product of the dimensions of the three sides such that they multiply to give x^3 + 9x^2 + 6x – 16. While there is more than one set of numbers that does that, presumably the question just looks for the factors of the given expression. One would normally pick the largest of these three factors (x + 8) and call it the length.
The three roots will be the three dimensions of the box. The three roots will be factors of 16, i.e., ±1, ±2, ±4, or ±8. Test each root by using polynomial division.
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Somebody else here said "The three roots will be the three dimensions of the box", but that doesn't make any sense. We're told that this function is the prism's volume. How could a prism with non-zero dimensions give us a volume of 0?
The volume of a rectangular prism is always length times width times height. So the product of those has to equal x^3 + 9x^2 + 6x - 16. However, there are lots of measurements whose product might be equal to this. For example, it could be that the length is 1 unit, the width is 1 unit, and the height is x^3 + 9x^2 + 6x - 16 for some unknown x. You could factor it into (x-1)(x+2)(x+8), but again, the length doesn't have to necessarily be one of these. If this expression is a volume of the function, then it begs the question of what exactly x is supposed to measure.
If it is a rectangular prism the area is the product of the dimensions of the three sides such that they multiply to give x^3 + 9x^2 + 6x – 16. While there is more than one set of numbers that does that, presumably the question just looks for the factors of the given expression. One would normally pick the largest of these three factors (x + 8) and call it the length.
It's a bad question.
x³ + 9x² + 6x - 16 = 0
The three roots will be the three dimensions of the box. The three roots will be factors of 16, i.e., ±1, ±2, ±4, or ±8. Test each root by using polynomial division.
Test (x - 1): (x³ + 9x² + 6x - 16)/(x - 1) = x³ + 10x + 16
Therefore, (x - 1) is the first root. The roots of x³ + 10x + 16 are:
x³ + 10x + 16 = (x + 2)(x + 8)
Thus, the three roots are: x = 1, x = -2, and x = -8.
(Note that the roots should not be negative numbers for the box to have a valid volume; it is my opinion that the only valid root is x = 1.)